The present invention relates to methods for determining and designing optical elements such as lenses and mirrors, in general, and to a method for determining and designing the topography of optical elements, in particular.
Methods for determining and designing optical elements, such as lenses and mirrors, are known in the art. These methods attempt to provide accurate surfaces for uni-focal optical elements and near accurate surfaces for multi-focal optical elements. Multi-focal lenses and mirrors provide a plurality of focal points, each for a different area in the optical element.
When designing an optical element, the designer determines a list of requirements which restrict the final result. Such requirements can include a general geometry of the requested surface, a collection of optical paths which have to be implemented in the requested surface and the like. Such optical paths can be determined in theory or as measurements of the paths of light rays, which include a plurality of rays, through the optical element.
According to one method, which is known in the art, an optical surface is determined according to a preliminary given surface, which is characterized by a finite number of parameters. The method calculates an optimal choice of the parameters, thereby determining the desired optical surface. The representation of the calculated optical surface can be given by polynomials or other known special functions.
It will be appreciated by those skilled in the art that such optical surfaces are limited in that the optimization is obtained according to a limited and finite number of parameters, while the number of restrictions can be significantly larger.
G. H. Guilino, xe2x80x9cDesign Philosophy For Progressive Addition Lensesxe2x80x9d, Applied Optics, vol. 32, pp. 111-117, 1993, provides a thorough survey of the methods for design principles of multi-focal lenses.
U.S. Pat. No. 4,315,673, to Guilino et al. is directed to a progressive power ophthalmic lens. Guilino describes a specific geometry, which utilizes specific functions to achieve an optical surface with varying power.
U.S. Pat. No. 4,606,622 to Fueter et al. is directed to a multi-focal spectacle lens with a dioptric power varying progressively between different zones of vision. Fueter describes a method for determining a surface according to a plurality of points. The method defines a twice continuously differentiable surface through these points. The surface is selected so as to achieve a varying optical surface power.
It is common practice in the industry to design optical elements consisting of several lenses (and thus several refractive surfaces). The designer of a complex optical element such as a camera has a finite and predetermined set of lenses at his disposal. The designer imposes requirements on the performance of the optical element, and then uses software to optimize the performance by choosing an optimal choice of lenses from the predetermined set. It will be appreciated that this approach limits the design of the optical element to the large but finite number of combinations of the predetermined set of lenses.
U.S. Pat. No. 5,581,347 to Le Saux et al. is directed to a method and device for measurement of an optical surface. A surface in the optical element is illuminated by light with a known wavefront. The slopes of the wavefront after reflection from or refraction by the surface are measured. An iterative process, starting from a given initial surface, attempts to find a surface that minimizes a predetermined merit function.
The present invention provides a novel method for determining surfaces of optical elements, which overcome the disadvantages of the prior art.
A method for determining the topography of at least one unknown surface of an optical element, the method including the steps of measuring geometrical properties, determining a set of integration equations from the geometrical properties and from the optical index or indexes of the optical element, and determining the topography of the at least one unknown surface from the set of equations. The geometrical properties measured are of a plurality of rays incident upon the optical element and of a corresponding plurality of rays affected by at least the at least one unknown surface.
There is also provided in accordance with a preferred embodiment of the present invention a method for designing an optical element having at least one unknown surface. The method includes the steps of prescribing geometrical properties, determining a set of integration equations from the geometrical properties and from the optical index or indexes of the optical element, and determining the topography of the at least one unknown surface from the set of equations. The geometrical properties prescribed are of a plurality of rays incident upon the optical element and of a corresponding plurality of rays affected by at least the at least one unknown surface.
Moreover, in accordance with a preferred embodiment of the present invention, the affected rays are refracted by at least one of the at least one unknown surface.
Alternatively, in accordance with a preferred embodiment of the present invention, the affected rays are reflected by at least one of the at least one unknown surface.
Furthermore, in accordance with a preferred embodiment of the present invention, the step of determining the topography includes the step of detecting whether the set of integration equations is solvable. When the set of integration equations is solvable, the set of integration equations is integrated, thereby determining the topography of the at least one surface. When the set of integration equations is not solvable, a function and auxiliary conditions for the set of integration equations are determined, and the function is optimized subject to the auxiliary conditions, thereby determining the topography of the at least one surface.
There is also provided in accordance with a preferred embodiment of the present invention a method for determining the topography of two unknown surfaces of an optical element. The method includes the step of measuring locations and directions of a plurality of rays incident upon the optical element and of a corresponding plurality of rays affected by at least the two unknown surfaces. The method also includes the steps of determining a set of integration equations from the locations and directions and from the optical index or indexes of the optical element, and determining the topography of the two unknown surfaces from the set of equations.
There is also provided in accordance with a preferred embodiment of the present invention a method for designing an optical element having two unknown surfaces. The method includes the step of prescribing locations and directions of a plurality of rays incident upon the optical element and of a corresponding plurality of rays affected by at least the two unknown surfaces. The method also includes the steps of determining a set of integration equations from the locations and directions and from the optical index or indexes of the optical element, and determining the topography of the two unknown surfaces from the set of equations.
Moreover, in accordance with a preferred embodiment of the present invention, the affected rays are refracted by at least one of the two unknown surfaces.
Alternatively, in accordance with a preferred embodiment of the present invention, the affected rays are reflected by at least one of the two unknown surfaces.
Furthermore, in accordance with a preferred embodiment of the present invention, the step of determining the topography includes the step of detecting whether the set of integration equations is solvable. When the set of integration equations is solvable, the set of integration equations is integrated, thereby determining the topography of the two unknown surfaces. When the set of integration equations is not solvable, a function and auxiliary conditions for the set of integration equations are determined, and the function is optimized subject to the auxiliary conditions, thereby determining the topography of the two unknown surfaces.
Moreover, in accordance with a preferred embodiment of the present invention, the method further includes step of determining a first reference plane from the locations and directions, wherein the integration equations include             f      x        =                                        u            1                    +                                    (                              f                -                                  h                  1                                            )                        ⁡                          [                                                                                          u                      1                                        ⁡                                          (                                              α                        /                        γ                                            )                                                        x                                +                                                                            u                      2                                        ⁡                                          (                                              β                        /                        γ                                            )                                                        x                                            ]                                                1          -                      (                                                            u                  1                                ⁢                                  α                  /                  γ                                            +                                                u                  2                                ⁢                                  β                  /                  γ                                                      )                              ≡      e1                  f      y        =                                        u            2                    +                                    (                              f                -                                  h                  1                                            )                        ⁡                          [                                                                                          u                      1                                        ⁡                                          (                                              α                        /                        γ                                            )                                                        y                                +                                                                            u                      2                                        ⁡                                          (                                              β                        /                        γ                                            )                                                        y                                            ]                                                1          -                      (                                                            u                  1                                ⁢                                  α                  /                  γ                                            +                                                u                  2                                ⁢                                  β                  /                  γ                                                      )                              ≡      e2                  g      x        =                                                      v              1                        ⁢                          ξ              x                                +                                    v              2                        ⁢                          η              x                                +                                    (                              g                -                                  h                  2                                            )                        ⁡                          [                                                                                          v                      1                                        ⁡                                          (                                              a                        /                        c                                            )                                                        x                                +                                                                            v                      2                                        ⁡                                          (                                              b                        /                        c                                            )                                                        x                                            ]                                                1          -                      (                                                            v                  1                                ⁢                                  a                  /                  c                                            +                                                v                  2                                ⁢                                  b                  /                  c                                                      )                              ≡      e3                  g      y        =                                                      v              1                        ⁢                          ξ              y                                +                                    v              2                        ⁢                          η              y                                +                                    (                              g                -                                  h                  2                                            )                        ⁡                          [                                                                                          v                      1                                        ⁡                                          (                                              a                        /                        c                                            )                                                        y                                +                                                                            v                      2                                        ⁡                                          (                                              b                        /                        c                                            )                                                        y                                            ]                                                1          -                      (                                                            v                  1                                ⁢                                  a                  /                  c                                            +                                                v                  2                                ⁢                                  b                  /                  c                                                      )                              ≡              e4        .                  
                ⁢        wherein                                u        1            =                                                  n              2                        ⁢            dx                    -                                    n              1                        ⁢            α                                                              n              1                        ⁢            γ                    -                                    n              2                        ⁢            dz                                ,          xe2x80x83        ⁢                  u        2            =                                                  n              2                        ⁢            dy                    -                                    n              1                        ⁢            β                                                              n              1                        ⁢            γ                    -                                    n              2                        ⁢            dz                                ,          xe2x80x83        ⁢                  v        1            =                                                  n              2                        ⁢            dx                    -                                    n              3                        ⁢            a                                                              n              3                        ⁢            c                    -                                    n              2                        ⁢            dz                                ,          xe2x80x83        ⁢    and                      v        2            =                                                  n              2                        ⁢            dy                    -                                    n              3                        ⁢            b                                                              n              3                        ⁢            c                    -                                    n              2                        ⁢            dz                                ;                      (                  dx          ,          dy          ,          dz                )            =                        1          D                ⁢                  (                      Dx            ,            Dy            ,            Dz                    )                      ,          
        ⁢          Dx      =              ξ        -        x        +                              a            c                    ⁢                      (                          g              -                              h                2                                      )                          -                              α            γ                    ⁢                      (                          f              -                              h                1                                      )                                ,          
        ⁢          Dy      =              η        -        y        +                              b            c                    ⁢                      (                          g              -                              h                2                                      )                          -                              β            γ                    ⁢                      (                          f              -                              h                1                                      )                                ,          
        ⁢                  Dz        =                              g            -                          f              ⁢                              xe2x80x83                            ⁢              and              ⁢                              xe2x80x83                            ⁢              D                                =                                                    Dx                2                            +                              Dy                2                            +                              Dz                2                                                        ;      
h1 denotes the height of a second reference plane above the first reference plane;
h2 denotes the height of a third reference plane above the first reference plane;
x and y denote the geometrical location of each of the incident rays on the second reference plane;
"xgr" and xcex7 denote the geometrical location of each of the affected rays on the third reference plane;
n1 denotes the optical index with respect to the incident rays;
n2 denotes the optical index between the two unknown surfaces;
n3 denotes the optical index with respect to the affected rays;
f denotes the height of one of the two unknown surfaces above the first reference plane;
g denotes the height of another of the two unknown surfaces above the first reference plane;
(xcex1, xcex2, xcex3) denotes the direction vector of each of the incident rays; and
(a, b, c) denotes the direction vector of each of the affected rays.
Furthermore, in accordance with a preferred embodiment of the present invention, the function is determined as
E=∫[w1(fxxe2x88x92e1)2+w2(fyxe2x88x92e2)2+w3(gxxe2x88x92e3)2+w4(gyxe2x88x92e4)2]dxdy,
and wi(x, y, f, g, ∇f, ∇g), i=1,2,3,4 are general weight functions.
There is also provided in accordance with a preferred embodiment of the present invention an optical element having at least one surface, the surface having a topography which is determined by the design method described hereinabove.
There is also provided in accordance with a preferred embodiment of the present invention a mold for producing an optical element having at least one surface, the surface having a topography which is determined by the design method described hereinabove.
There is also provided in accordance with a preferred embodiment of the present invention a mold for producing an optical element, the mold having a surface whose topography is determined by the design method described hereinabove.
There is also provided in accordance with a preferred embodiment of the present invention a measuring device for determining the topography of two unknown surfaces of an optical element. The measuring device includes a first optical system for measuring locations and directions of rays incident on the optical element, a second optical system for measuring locations and directions of rays affected at least by the two unknown surfaces, and software means for determining the topography of the two unknown surfaces from the locations and directions and from optical properties of the optical element.
There is also provided in accordance with a preferred embodiment of the present invention a measuring device for determining the topography of an unknown surface of an optical element. The measuring device includes a first optical system for measuring either of locations and directions of rays incident on the optical element, a second optical system for measuring locations and directions of rays affected at least by the unknown surface, and software means for determining the topography of the unknown surface from a set of integration equations, the set of integration equations determined from the locations and directions and from optical properties of the optical element.
There is also provided in accordance with a preferred embodiment of the present invention a measuring device for determining the topography of an unknown surface of an optical element. The measuring device includes a first optical system for measuring locations and directions of rays incident on the optical element, a second optical system for measuring either of locations and directions of rays affected at least by the unknown surface, and software means for determining the topography of the unknown surface from a set of integration equations, the set of integration equations determined from the locations and directions and from optical properties of the optical element.